Proof that chromatic number is $< 9$

Question

Let $G= (V,E)$ be a graph such that:
$E = E_1 \cup E_2$
$G_1 = (V, E_1)$ is planar
$G_2 = (V, E_2)$ is forest
Proof that chromatic number is $< 9$

My observations

Each planar graph has chromatic number $\le 4$. Also each forest has chromatic number $2$ if $|V| > 1$

But how can I connect that and have convince about chromatic number $<9$?

Answer 1

Given colorings $C_1$ of $G_1$ and $C_2$ of $G_2$, construct a new coloring $C_1 \times C_2$ where $C_1 \times C_2(v) = ( C_1(v),C_2(v))$. This is a valid coloring of the graph (check this!). Since there are at most 4 colors in $C_1$ and at most 2 in $C_2$, there are at most 8 in $C_1 \times C_2$.